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Topological structure and dynamicsof three-dimensional active nematicsGuillaume Duclos1*, Raymond Adkins2*, Debarghya Banerjee3,4, Matthew S. E. Peterson1,Minu Varghese1, Itamar Kolvin2, Arvind Baskaran1, Robert A. Pelcovits5, Thomas R. Powers6,5,Aparna Baskaran1, Federico Toschi7,8, Michael F. Hagan1, Sebastian J. Streichan2,Vincenzo Vitelli9, Daniel A. Beller10†, Zvonimir Dogic1,2†

Topological structures are effective descriptors of the nonequilibrium dynamics of diverse many-bodysystems. For example, motile, point-like topological defects capture the salient features oftwo-dimensional active liquid crystals composed of energy-consuming anisotropic units. We dispersedforce-generating microtubule bundles in a passive colloidal liquid crystal to form a three-dimensionalactive nematic. Light-sheet microscopy revealed the temporal evolution of the millimeter-scalestructure of these active nematics with single-bundle resolution. The primary topological excitationsare extended, charge-neutral disclination loops that undergo complex dynamics and recombinationevents. Our work suggests a framework for analyzing the nonequilibrium dynamics of bulkanisotropic systems as diverse as driven complex fluids, active metamaterials, biological tissues,and collections of robots or organisms.

The sinuous change in the orientation ofbirds flocking is a common but startlingsight. Even if one can track the orienta-tion of each bird, making sense of suchlarge datasets is difficult. Similar chal-

lenges arise in disparate contexts from mag-netohydrodynamics (1) to turbulent culturesof elongated cells (2), where oriented fieldscoupled to velocity undergo complex dynam-ics. Tomake progress with such extensive three-dimensional (3D) data, it is useful to identifyeffective degrees of freedom that allow a coarse-grained description of the collective nonequi-librium phenomena. Promising candidatesare singular field configurations locally pro-tected by topological rules (3–9). Examplesof such singularities in 2D are the topolog-ical defects that appear at the north and southpoles when covering the Earth’s surface withparallel lines of longitude or latitude. Thesepoint defects are characterized by the windingnumber of the corresponding orientation field.The quintessential systems with orienta-

tional order are nematic liquid crystals, which

are fluids composed of anisotropic molecules.In equilibrium, nematics tend to minimizeenergy by uniformly aligning their anisotropicconstituents, which annihilates topological de-fects. By contrast, in active nematic materials,which are internally driven away from equi-librium, the continual injection of energy de-stabilizes defect-free alignment (10, 11). Theresulting chaotic dynamics are effectively rep-resented in 2D by point-like topological de-fects that behave as self-propelled particles(12–16). The defect-driven dynamics of 2Dactive nematics have been observed in manysystems ranging frommillimeter-sized shakengranular rods and micrometer-sized motilebiological cells to nanoscale motor-driven bio-logical filaments (17–23). Several obstacles havehindered generalizing topological dynamicsof active nematics to 3D. The higher dimen-sionality expands the space of possible defectconfigurations. Discriminating between differ-ent defect types requires measurement of thespatiotemporal evolution of the director fieldon macroscopic scales using materials thatcan be rendered active away from surfaces.The 3D active nematics that we assembled

are based on microtubules and kinesin mo-lecular motors. In the presence of a depletingagent, these components assemble into iso-tropic active fluids that exhibit persistentspontaneous flows (17). Replacing a broadlyacting depletant with a specific microtubulecross-linker, PRC1-NS, enabled assembly of acomposite mixture of low-density extensilemicrotubule bundles (~0.1% volume frac-tion) and a passive colloidal nematic basedon filamentous viruses (Fig. 1A), a strategythat is similar to work on the living liquid

crystal (21). Adenosine 5′-triphosphate (ATP)–fueled stepping of kinesin motors generatesmicrotubule bundle extension and activestresses that drive the chaotic dynamics ofthe entire system (movie S1). Birefringence ofthe compositematerial indicates local nematicorder (Fig. 1B), in contrast to active fluids lack-ing the passive liquid crystal component.Elucidating the spatial structure of a 3D

active nematic requires measurement of thenematic director field on scales from micro-meters to millimeters. Furthermore, uncov-ering its dynamics requires acquisition ofthe director field with high temporal resolu-tion. To overcome these constraints, we useda multiview light sheet microscope (Fig. 1C)(24). The spatiotemporal evolution of the ne-matic director field n(x,y,z,t) was extractedfrom a stack of fluorescent images using thestructure tensor method. Spatial gradientsof the director field identified regions withlarge elastic distortions (Fig. 1D and movieS2). Three-dimensional reconstruction of suchmaps revealed that large elastic distortionsmainly formed curvilinear structures, whichcould either be isolated loops or belong to acomplex network of system-spanning lines(Fig. 1E and movie S3). These curvilinear dis-tortions are topological disclination linescharacteristic of 3D nematics. Similar struc-tures were observed in numerical simulationsof 3D active nematic dynamics using either ahybrid lattice Boltzmann method or a finitedifference Stokes solver numerical approach(Fig. 1F) (25, 26).Reducing the ATP concentration slowed

down the chaotic flows, which revealed thetemporal dynamics of the nematic directorfield. In turn, this identified the basic eventsgoverning the dynamics of disclination lines(movie S4).We focused on characterizing theclosed loop disclinations because they are theobjects seen to arise or annihilate in the bulk.Isolated loops nucleated and grew from un-distorted, uniformly aligned regions (Fig. 2A,figs. S1 and S2, and movie S5). Likewise, loopsalso contracted and self-annihilated, leavingbehind a uniform region (Fig. 2B, figs. S1 andS2, and movie S6). Furthermore, expand-ing loops frequently encountered and subse-quently merged with the system-spanningnetwork of distortion lines, whereas the dis-tortion lines in the network self-intersectedand reconnected to emit a new isolated loop(Fig. 2, C and D; figs. S1 and S2; and movies S7and S8).Topological constraints require that topo-

logical defects can only be created in setsthat are, collectively, topologically neutral.Point-like defects in 2D active nematics thusalways nucleate as pairs of opposite windingnumber (13). In 3D active nematics, an iso-lated disclination loop as a whole has twotopological possibilities: It can either carry a

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1Department of Physics, Brandeis University, Waltham, MA02453, USA. 2Department of Physics, University of California,Santa Barbara, CA 93111, USA. 3Max Planck Institute forDynamics and Self-Organization, 37077 Göttingen, Germany.4Instituut-Lorentz, Universiteit Leiden, 2300 RA Leiden,Netherlands. 5Department of Physics, Brown University,Providence, RI 02912, USA. 6School of Engineering, BrownUniversity, Providence, RI 02912, USA. 7Department of AppliedPhysics, Eindhoven University of Technology, 5600 MBEindhoven, Netherlands. 8Instituto per le Applicazioni delCalcolo CNR, 00185 Rome, Italy. 9James Frank Institute andDepartment of Physics, The University of Chicago, Chicago,IL 60637, USA. 10Department of Physics, University ofCalifornia, Merced, CA 95343, USA.*These authors contributed equally to this work.†Corresponding author. Email: dbeller@ucmerced.edu (D.A.B.);zdogic@ucsb.edu (Z.D.)

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microtubulefd virus

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Fig. 1. Assembling 3D active nematics and imaging their director field. (A) Schematic of the 3D activenematic system: active stress–generating extensile microtubule bundles are dispersed in a passive colloidalliquid crystal. (B) Active 3D nematic imaged with widefield fluorescent microscopy (left) and polarizedmicroscopy (right). Birefringence indicates local nematic order. (C) Multiview light sheet microscopy allowsfor 3D imaging of millimeter-sized samples with single-bundle resolution. (D) Left: A 2D slice of fluorescentmicrotubule bundles with highlighted elastic distortions. Right: Corresponding elastic distortion energy map, withan overlaid nematic director field (red). (E) Three-dimensional elastic distortion map revealing the presenceof curvilinear rather than point-like singularities. An entangled network of lines coexists with isolated loops.(F) Hybrid lattice Boltzmann simulations yield a similar structure of 3D active nematics. All experimentalsamples consist of passive fd viruses at 25 mg/mL and microtubules at 1.33 mg/mL.

Fig. 2. Dynamics of experimentallyobserved disclination loops. (A) Loopnucleation from a defect-free region.(B) Loop self-annihilation leavesbehind a defect-free nematic.(C) Disclination line self-intersects,reconnects, and emits a loop.(D) Disclination loop intersects,reconnects, and merges with adisclination line. Each boundingbox is 30 × 30 × 38 mm. The timeinterval between two pictures is 12 s.

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monopole charge or be topologically neutral,depending on its director winding structure.Because charged topological loops can onlyappear in pairs, nucleation of isolated loopsas observed in our system implies their to-pological neutrality.

To establish that the closed-loop distor-tions are nematic disclination loops with nonet charge, we characterized their topolog-ical structure. In 2D nematics, point-like dis-clination defects are characterized by thewinding number or topological charge(s). The

lowest-energy disclinations have s = ±1/2,which corresponds to a p rotation of the di-rector field in the same sense or the oppo-site sense, respectively, as the traversal ofany closed path encircling only the defectof interest. In 3D nematics, point-like defects

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Fig. 3. Structure of disclinations lines, wedge-twist, and pure-twist loops. (A) Disclination linewhere a local +1/2 wedge winding continuouslytransforms into a –1/2 wedge through an interme-diate twist winding. The director field winds by pabout the rotation vector W (black arrows), whichmakes angle b with the tangent t (orange arrow) andis orthogonal to the director field everywhere ineach slice. For ±1/2 wedge windings, b = 0 and p.b = p/2 indicates local twist winding. Referencedirector no (brown) is held fixed. Color map indicatesangle b. (B) Wedge-twist loop where local winding asreflected by angle b varies along the loop. W isspatially uniform and forms an angle g = p/2 withthe loop’s normal, N. The winding in the fourillustrated planes corresponds to the profiles of thesame colors shown in (A), with dashed edges ofsquares aligned to match the local director field.Double-headed brown arrows indicate nout, thedirector just outside the loop. (C) Pure-twist loop,with W both uniformly parallel to loop normal N(g = 0) and perpendicular to the tangent vector.

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Fig. 4. Structure of disclination loops in experiments and theory. (A) Twoorthogonal views of an experimental wedge-twist loop overlaid onto a fluo-rescent image of the microtubules. The nematic director is shown in red.(B and E) Structure of wedge-twist disclination loops in experiments and sim-ulation. (C and F) Structure of pure-twist disclination loops from experimentand simulation. Panels show the director field’s winding in the correspondingcross-sections on the experimental loops. (D) Distribution of loop types

extracted from experiment (N = 268) and hybrid lattice Boltzmann simulations(N = 94). |cos(g)| = 0 for wedge-twist loops and 1 for pure-twist loops.Distributions of standard deviations of |cos(g)| are shown in fig. S3. The countof simulated loops includes analysis of some loops at multiple time pointsbecause we did not track loop identity in the complex flow dynamics. Coloringof loops indicates the angle b. Scales and bounding boxes for the loops areshown in fig. S4.

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from 2D systems are generalized to discli-nation lines, where the director similarlyhas a p winding, affording a broader varietyof director configurations. We define t to bethe disclination line’s local tangent unitvector. The director field winds by p abouta direction specified by W, the rotation vec-tor, which can make an arbitrary angle bwith t (27). If W points antiparallel or par-allel to t, then the local director field rotatesin the plane orthogonal to t, assuming thedisclination profiles familiar from 2D nem-atics. These configurations in which b isequal to 0 or p are said to have a local wedgewinding (Fig. 3A). IfW is perpendicular to t,then b = p/2 and the director forms a spa-tially varying angle away from the orthog-onal plane, locally creating what is called atwist winding. Because Wmay point in anydirection relative to t, both W and b canvary continuously along a disclination line(movie S9).For disclination lines forming loops, W

can vary continuously providing it returnsto its original orientation upon closure, lead-ing to a broad range of possible windingvariations. A family of loops of particularrelevance to 3D active nematics is character-ized by a spatially uniform W, interpolatingbetween two emblematic geometries: wedge-twist and pure-twist loops. In the wedge-twist loop, W makes an angle g = p/2 with theloop normal N (Fig. 3B). As the disclination’stangent t rotates by 2p upon traveling aroundthe loop, the angle b varies from 0 (+1/2 wedge)to p/2 (twist), to p (–1/2 wedge), then backto p/2, and finally returning to 0 (movie S9)(27, 28). The pure-twist loop has W uniformlyparallel to N, so g = 0 and W is perpendic-ular to t (b = p/2, twist profile) at all pointson the loop (Fig. 3C) (27, 29). In this familyof loops, the director just outside the loop,nout, is also uniform. The lack of winding ofboth W and nout implies that both wedge-twist and pure-twist loops are topologicallyneutral (30, 31).Experimental measurements of the director

field allowed us to fully characterize the topo-logical structure of the disclination loops(Fig. 4). Analysis of the director field indicatedthat the distortion lines and loops have the pwinding indicative of disclinations (Fig. 1, Eand F), with continuous variation of b, whichindicates local winding. Furthermore, most ofthe analyzed loops were well approximated bythe family of curves where W and nout variedlittle along the loop circumferences. Categoriz-ing loops according to their g values revealedthat the entire continuous family from wedge-twist (Fig. 4, A and B) to pure-twist (Fig. 4C)was represented, with the latter being moreprevalent (Fig. 4D). Structural analysis re-vealed topological neutrality, as all 268 ex-perimental loops and all 94 loops extracted

from hybrid lattice Boltzmann simulationscarried no charge. This demonstrates thatamong many possible configurations, topo-logically neutral loops are the dominant ex-citation mode of 3D active nematics. Thesame class of loop geometries also domi-nated the dynamics in our numerical sim-ulations of bulk 3D active nematics and inconfined active nematics (Fig. 4, E and F)(25, 26, 32, 33). The phenomenology observedis a direct consequence of activity-inducedflows and is insensitive to backflows inducedby reactive stresses. This conclusion is sup-ported by the agreement of results from themechanical model considered in the hybridlattice Boltzmann method and the purely kin-ematic Stokes method.In 2D active nematics, self-amplifying bend

distortions give rise to the nucleation of apair of topological defects of opposite charge(12–18). Nucleation of isolated, topologicallyneutral wedge-twist loops are the 3D analog ofthe 2D defect-creation process. Specifically, across-section through the +1/2 and –1/2 wedgeprofiles recalls unbinding of a pair of pointdisclinations in 2D (Fig. 5, A and B). The +1/2wedge profile typically appears on the side ofthe growing bend distortion, oriented awayfrom the –1/2 wedge profile. Similarly, wedge-twist loops with the +1/2 wedge profile ori-ented inward toward the –1/2 wedge aredriven to shrink by active and passive stresses.Unlike in 2D active nematics, after nucleation,

thewedge profiles remain bound to each otherthrough a disclination loop that includespoints with a local twist winding. It is possiblethat some analyzed pure-twist loops haveevolved from wedge-twist loops by continu-ous deformation of local winding character.However, both simulations and experimentsshowed cases of loop nucleation in nearlypure-twist (g ≈ 0) geometries from previ-ously defect-free regions. Local active nematicstresses alone are not expected to drive growthof a pure-twist loop (Fig. 5D). One possibilityis that long-range hydrodynamic flows buildup twist distortions that locally relax throughcreation of a pure-twist loop (Fig. 5C andmovie S10).By coupling a flow field to an orientational

order parameter with curvilinear topologicaldefects, 3D active nematics display dynamicseven more complex than the chaotic flows of2D active systems. Combined with emergingtheoretical work (32, 33), the experimentalmodel system described herein offers a plat-form with which to investigate the role of to-pology, dimensionality, and material orderin the chaotic internally driven flows of ac-tive soft matter. Furthermore, the use of amultiview light sheet imaging technique dem-onstrates its potential to unravel dynamicalprocesses in diverse nonequilibrium soft ma-terials, such as relaxation of nematic liquidcrystals upon a quench or their deformationunder external shear flow (3, 34).

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Fig. 5. Nucleation mechanism of wedge-twist and pure-twist loops. (A) Nucleation and growth of awedge-twist disclination loop through a self-amplifying bend distortion. Purple rods represent the 2D directorfield through the local ±1/2 wedge profiles. (B) Schematic of a wedge-twist loop and the director field inthe plane that intersects ±1/2 wedge profiles. (C) Pure-twist disclination loop nucleates and grows froma local twist distortion (movie S10). Black arrows indicate the local buildup of the twist distortion.Insert shows the top view of a growing twist disclination loop. (D) Schematic of a pure-twist loop andthe director field in the loop’s plane.

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ACKNOWLEDGMENTS

We thank B. Lemma, R. Subramanian, and M. Ridilla for help inprotein purification. D.A.B. acknowledges S. Čopar, S. Žumer, andM. Ravnik for helpful discussions on disclinations in 3D activenematics. Funding: Experimental work was supported by the U.S.Department of Energy, Office of Basic Energy Sciences, throughaward DE-SC0019733 (G.D., R.A., I.K., and Z.D.). G.D. and I.K.

also acknowledge support from HFSP fellowships. R.A.acknowledges support from NSF-GRFP. Theoretical modeling wassupported by NSF-DMR-1855914, NSF-MRSEC-1420382, andNSF-CBET-1437195 (D.A.B., M.P., M.V., Ar.B., Ap.B., M.F.H., R.A.P.,and T.R.P.). Computational resources were provided by the NSFthrough XSEDE computing resources (MCB090163) and TU/ethrough the F&F computing cluster. We also acknowledge use ofthe Brandeis optical, HPCC, and biosynthesis facilities supportedby NSF-MRSEC-1420382. D.B. was supported by FOM and NWO.V.V. was supported by the Army Research Office under grantW911NF-19-1-0268 and NSF-MRSEC (DMR-1420709). S.J.S. wassupported by NIH-R00 award 5R00HD088708-05. D.A.B. thanksthe Isaac Newton Institute for Mathematical Sciences for supportduring the program “The Mathematical Design of New Materials,”supported by EPSRC grant EP/R014604/1. Author contributions:G.D., I.K., S.J.S., and R.A. conducted experimental research;G.D., I.K., S.J.S., R.A., M.S.P., and D.A.B. analyzed experimentaland theoretical data. D.B., F.T., and V.V. developed hybrid latticeBoltzmann simulations. M.S.P., M.V., Ar.B., Ap.B., and M.F.H.developed and applied finite difference Stokes solver code. D.A.B.,R.P., and T.R.P. conducted theoretical modeling and interpretationof data. G.D., V.V., Ap.B., M.F.H., D.A.B., and Z.D conceived thework. G.D., D.A.B., V.V., and Z.D. wrote the manuscript. All authorsreviewed the manuscript. Competing interests: The authorsdeclare no competing interests. Data and materials availability:Experimental director field data, the code to detect and analyze thetopological loops, and the Stokes solver are available on Dryad(25). Lattice Boltzmann code and director generated by this codeare available on REPOSITORY (26).

SUPPLEMENTARY MATERIALS

science.sciencemag.org/content/367/6482/1120/suppl/DC1Materials and MethodsSupplementary TextFigs. S1 to S4Captions for Movies S1 to S10References (35–53)

16 September 2019; accepted 7 February 202010.1126/science.aaz4547

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Topological structure and dynamics of three-dimensional active nematics

Streichan, Vincenzo Vitelli, Daniel A. Beller and Zvonimir DogicBaskaran, Robert A. Pelcovits, Thomas R. Powers, Aparna Baskaran, Federico Toschi, Michael F. Hagan, Sebastian J. Guillaume Duclos, Raymond Adkins, Debarghya Banerjee, Matthew S. E. Peterson, Minu Varghese, Itamar Kolvin, Arvind

DOI: 10.1126/science.aaz4547 (6482), 1120-1124.367Science 

, this issue p. 1120; see also p. 1075Sciencecollapse of spatially extended topological defects in three dimensions.Perspective by Bartolo). This setup makes it possible to directly watch the nucleation, deformation, recombination, and light-sheet microscopy to watch the motion of nematic molecules driven by the motion of microtubule bundles (see theexperimental visualization of the structure and dynamics of disclination loops in active, three-dimensional nematics using

report theet al.materials or through mesoscale simulations of the local orientation of the molecules. Duclos Orientational topological defects in liquid crystals, known as disclinations, have been visualized in polymeric

Watching defects flow and grow

ARTICLE TOOLS http://science.sciencemag.org/content/367/6482/1120

MATERIALSSUPPLEMENTARY http://science.sciencemag.org/content/suppl/2020/03/04/367.6482.1120.DC1

CONTENTRELATED http://science.sciencemag.org/content/sci/367/6482/1075.full

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